| dc.description.abstract | Discrete Fourier Transform (DFT) is a widely used tool in the field of electrical engineering. In the name of Mattson-Solomon polynomials, DFT was used in the context of linear cyclic codes in the early days of coding theory. From then on, it has become a very useful tool for investigating structural properties of many different families of codes over different alphabets.
Codes with rich algebraic structure are of strong interest to coding theorists due to the ease of design and decoding. Classical families of linear cyclic codes like BCH codes and Reed-Muller codes were the center of attraction for a long time. Though rich algebraic structures like linearity and cyclicity make design and decoding easier, they restrict the freedom of choice for long good codes. So, successful attempts have been made to slacken the restrictions of linearity and cyclicity to look for good codes. However, to keep the problem of designing and decoding tractable, neither of the structural restrictions is completely given away.
Slackening the restriction of cyclicity gives the class of quasi-cyclic codes. Refining the works of Chen, Peterson, and Weldon [1], Kasami [2] proved that 2-quasi-cyclic codes asymptotically meet a slightly loose version of the Gilbert-Varshamov bound. Many quasi-cyclic codes were found which are best known codes of their lengths. The structural properties and enumeration of quasi-cyclic codes were discussed using different approaches in [3–5]. The algebraic structure of ?-quasi-cyclic codes of length n is investigated in a part of this thesis with the help of the conventional DFT of length n. A way is given to derive a lower bound on the minimum Hamming distance of a quasi-cyclic code using the DFT domain characterization of the code.
Compromising linearity gives codes which are linear over some subfield F? of the alphabet field F_{q^m} but not necessarily linear over F_{q^m}. Different good classes of F?-linear cyclic codes (F?LC) over F_{q^m} like twisted BCH codes and subspace subcodes of Reed-Solomon (SSRS) codes were found by different authors [6, 7]. A part of this thesis characterizes the F?-linear cyclic codes over F_{q^m} in the DFT domain when the length is relatively prime to q. With respect to any given F?-basis of F_{q^m}, every n-length F?-linear cyclic code over F_{q^m} can be considered as a linear m-quasi-cyclic code of length mn over F?. A method is given to derive a lower bound on the minimum Hamming distance of a quasi-cyclic code using the DFT domain characterization of the corresponding F?LC code.
The classes of codes like cyclic codes, abelian codes, quasi-cyclic codes, and quasi-abelian codes [8] are defined by certain restrictions on their permutation groups. All these classes of codes are defined to be with their permutation group containing a certain type of abelian subgroup. Given any abelian subgroup G of the permutation group of the coordinates such that the exponent is relatively prime to q, G-invariant codes are investigated with the help of a suitably defined DFT. Duals of G-invariant codes and self-dual G-invariant codes are characterized in the transform domain. A general formula for the enumeration of self-dual G-invariant codes is found using this characterization. A way to derive a lower bound on the minimum Hamming distance of a G-invariant code is outlined. Karlin’s decoding algorithm for a systematic quasi-cyclic code with a single row of circulants in the generator matrix is extended to the case of systematic quasi-abelian codes. In particular, this can be used to decode systematic quasi-cyclic codes with columns of parity circulants in the generator matrix.
Codes over integer residue rings and more generally over Galois rings have received serious attention [9–15] after it was shown [16] that some important families of nonlinear binary codes can be obtained by Gray map from linear codes over ??. A part of this thesis generalizes the transform domain study of G-invariant codes (G is as in the previous paragraph). Recently, Blackford and Ray-Chaudhuri [17] used a transform domain technique to permutation groups of cyclic codes over Galois rings. Here, their technique is extended to permutation groups of abelian codes over Galois rings.
A code is said to be affine invariant if it is invariant under the affine permutations. Often it is comparatively easier to determine the full permutation group of an affine invariant code [18–22]. The conditions for extended cyclic codes over finite fields and integer residue rings to be affine-invariant were derived by respectively Kasami, Lin, and Peterson [23] and Abdukhalikov [24]. Blackford and Ray-Chaudhuri [17] used a transform domain approach to characterize the affine invariant extended cyclic codes of length 2^m over the subrings of GR(4, m) and using this characterization, they found new classes of affine invariant BCH codes over Galois rings. In this thesis, their approach is extended to extended cyclic codes of length 2^m over any subring of GR(2^e, m) and also to extended cyclic codes of length p^m over GR(p², m) for arbitrary prime p. New classes of affine invariant codes are found using these results. | |
